Let $X$ and $Y$ be affine varieties over a field $k$, with associated finitely generated, integral $k$-algebras $A(X)$ and $A(Y)$. We want to show that every irreducible component of the fibre product $X\times Y$ has dimension $\dim X+\dim Y$, motivated by this question. In algebraic terms, we want to show that, for any minimal prime ideal $P_i$ in the tensor product $A(X)\otimes_k A(Y)$, the maximal length of a chain of prime ideals starting at $P_i$ is $\dim X+\dim Y$.
To do this, we tried/read the following: Let $N_X\subseteq A(X)$ and $N_Y\subseteq A(Y)$ be Noether normalizations of $A(X)$ resp. $A(Y)$. We can prove that, in this case, $N_X\otimes_k N_Y$ is a a free $K$-algebra and furthermore a Noether normalization of $A(X)\otimes_k A(Y)$ (and has Krull dimension $\dim X + \dim Y$). If every minimal prime ideal $P_i$ in $A(X)\otimes_k A(Y)$ lies over $0\otimes 0$ in that Noether normalization, we can use Going up and Incomparability to prove that every maximal chain of prime ideals starting at $P_i$ has length $\dim X + \dim Y$, as done e.g. here, lemma 11.8.
So our question is: How to prove that every minimal prime ideal lies over $0$? The Local Algebra book by Serre claims this (on page 48), but we don't understand his reasoning and wonder if we are missing something very simple.